A registered result in its source context
Original ledger classification: Identity [I] (exact dual-vector + Sherman-Morrison + controls) + Physical [P] (bridge reading)
Recorded statement: THE DUAL ANCHOR LEMMA (the inverse flavor response of the anchor IS the Nariai root; the third purely algebraic leg of the flavor-horizon bridge; external review 2026-06-12, validated): (1) THE DUAL ANCHOR [I]: d := a^T R^-1 = a^T L^-1 =…
Statement, assumptions and sources
Paper context: Standard Model and flavor structure
The exact identifier, original classification and full claim are available below.
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FLAV.DUAL.01
Original ledger classification: Identity [I] (exact dual-vector + Sherman-Morrison + controls) + Physical [P] (bridge reading)
Original claim text
THE DUAL ANCHOR LEMMA (the inverse flavor response of the anchor IS the Nariai root; the third purely algebraic leg of the flavor-horizon bridge; external review 2026-06-12, validated): (1) THE DUAL ANCHOR [I]: d := a^T R^-1 = a^T L^-1 = (-1/2, -1/2, 1) with d.1 = 0, d.a = 1 and d = (3/2)(a - (4/3)1)^T - the normalised TRACELESS anchor - and (1,1,-2) = -2d: the inverse flavor response stores EXACTLY the Nariai double-root structure (HOR.NARIAI.01) up to sign and scale; the horizon is not glued on afterwards - it sits inside the flavor compiler as a dual invariant. (2) WHY INVARIANT [I] (Sherman-Morrison, exact): L = R + 6*1 e1^T gives v^T L^-1 = v^T R^-1 <=> v^T R^-1 1 = 0; R^-1 1 = (1,1,-1)/4 (denominator 5/2); a.(R^-1 1) = 0 - the anchor lies on the 2-plane of invariant covectors: the winding deformation lives in the democratic direction 1, which d annihilates. (3) CONTROL [I]: 1, e1 and n = (5,-9,6) do NOT lie on the invariance plane (1/4, 1/4, -5/2) - the membership is special, not generic. (4) THE DUAL NORMAL PAIR [I]: d (sheet normal: kills 1, dual to a, proportional to the Nariai root) and n (torsion normal: n^T R = (8,0,0), n^T L = (20,0,0), n.a = 8, n.1 = 2 = |Z2| cf. FLAV.DIAMOND.02) form the dual basis of the flavor boundary. (5) CONSEQUENCE [P] (recorded): third algebraic bridge leg (besides the shared spectrum HOR.CLOCK.04 and the entropy power law HOR.CLOCK.06); supports the reduced seam reading (HOR.CLOCK.10); the R4' selector programme (FLAV.H2.05) acquires a geometric target: derive the PAIR (d, n), not n alone
Original classification detail
Identity: a^T R^-1 = a^T L^-1 = Nariai root/(-2); invariance <=> orthogonality to R^-1 1; dual normal pair (d n); R4' target = the pair
Dependencies as recorded
HOR.NARIAI.01; FLAV.DIAMOND.02; FLAV.H2.05 (selectors); HOR.CLOCK.04; HOR.CLOCK.06
Open the original ledger source, row 219Location recorded in the ledger
These document locations are recorded directly on the ledger row. They are document level navigation, rather than a newly inferred section citation.
- Standard Model and flavor structure
- Horizon readouts
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- Front matter / supportResearch contracts
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- Front matter / supportStandard Model and flavor structure
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
- 2026-06-12Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Research Contract 1 — [formula in source]tfpt research contracts
All 9 linked sections
Registered verification modules
- v134_dual_anchor.pyRegistered verification module · registry
Read the description recorded in the module registry
THE DUAL ANCHOR LEMMA (the inverse flavor response of the anchor IS the Nariai root — the third purely algebraic leg of the flavor↔horizon bridge): d := aᵀR⁻¹ = aᵀL⁻¹ = (−1/2, −1/2, 1) with d·1 = 0, d·a = 1 and d = (3/2)(a − (4/3)1) — the normalised TRACELESS anchor — and (1,1,−2) = −2d: the inverse flavor response stores exactly the Nariai double-root structure (v101) up to sign and scale, so the horizon is not glued on afterwards — it sits inside the flavor compiler as a dual invariant [I]; WHY INVARIANT (Sherman–Morrison, exact): L = R + 6·1e₁ᵀ makes a covector winding-invariant iff it annihilates R⁻¹1 = (1,1,−1)/4, and the anchor does — the winding deformation lives in the democratic direction, which d kills [I]; CONTROL: 1, e₁ and n = (5,−9,6) do NOT lie on the invariance plane (1/4, 1/4, −5/2) — the membership is special, not generic [I]; THE DUAL NORMAL PAIR: d (sheet normal: kills 1, dual to a, ∝ Nariai root) and n (torsion normal: nᵀR = (8,0,0), nᵀL = (20,0,0), n·a = 8, n·1 = 2 = |Z₂|) form the dual basis of the flavor boundary [I]; consequence [P]: third algebraic bridge leg beside the shared clock spectrum (v126) and the entropy power law (v129); the R4′ selector programme acquires a geometric target — derive the PAIR (d, n), not n alone
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